An exact algorithm for the multi-period inspector scheduling problem
Introduction
The multi-period inspector scheduling problem (MPISP) was first introduced by Qin, Ming, Zhang, Xie, and Lim (2015). This problem originated in the practice of an international retailer’s buying office, based in Hong Kong. For retail business, the buying office regularly procures assorted products from thousands of suppliers that are mainly located in mainland China. As stipulated in the procurement contract, the manufacturing of these products must be inspected on site to achieve quality control. Therefore, when it is ready to manufacture the ordered products, each supplier sends the buying office an inspection request, which prescribes a workload, an inspection site, and a time window for the inspection starting time.
To meet the inspection requests from all the suppliers, the buying office dispatches a team of professional quality inspectors to perform on-site inspections of products. The buying office usually plans a weekly schedule for the inspectors, assigning each of them a certain number of inspection jobs. As a result, each inspector sets off from the buying office on Monday, visits a set of suppliers to perform inspections, and finally returns to the office on Friday or on an earlier workday if all his/her assigned inspections have been completed. In consideration of the actual working time of the suppliers’ factories and the regulated labor intensity for inspectors, the inspection and traveling (between different inspection sites) can only be conducted during the working hours of a day, e.g., 8 am to 6 pm. Therefore, after finishing some inspections in one working day, an inspector finds a hotel to take a rest overnight, and the hotel is near his/her last/next inspection site.
Fig. 1 shows an example of routing for two inspectors in a time horizon of five working days, where circle 0 represents the depot (the buying office), rectangles 1 to 14 represent assigned inspection jobs at fourteen different suppliers, and circle 1 indicates that supplier 1 was visited without carrying out an inspection (be used as an intermediate point before reaching the destination). Inspector A’s route is marked by red lines, while inspector B’s route is marked by blue lines. For example, the sub-routes traveled and jobs carried out by inspector B in five workdays are as follows: (1) sub-route , jobs 7 and 8; (2) sub-route , job 10; (3) sub-route , job 11; (4) sub-route , jobs 12 and 13; (5) sub-route , job 14. As we can see, a distinguishing feature of this problem is the segmentation of routing time by working periods. The inspectors are not required to return to the depot every day. Instead, they stay at one supplier for a rest at the end of each day. Particularly, on day 4, after inspector B finishes the inspection job at supplier 13, the remaining working time at that day is not sufficient for traveling from supplier 13 to supplier 14. Thus, inspector B travels to supplier 1 that is near to supplier 13, takes a rest overnight there, and travels to supplier 13 on the next day. As a result, supplier 1 has been visited by both inspectors, but only inspector A performs an inspection there. Therefore, each supplier can be visited more than once in this problem.
Since the number of inspectors is limited,1 it is hard for them to fulfill all inspections, and the unfulfilled inspection jobs (e.g., the inspection at supplier 9 in Fig. 1) are outsourced to some third-party companies at an extra cost. Consequently, the objective of the buying office is to find a set of routes for inspectors to perform inspections such that all the above practical constraints are satisfied, and the total workloads assigned to the inspectors are maximized (so that the inspection outsourcing cost is minimized).
As a real-life problem that stems from the practice, some other applications of the MPISP can be found in vehicle routing problems where the routing time is divided into different segments. For example, one application of the MPISP arises in freight distribution in many Chinese cities. In recent years, air pollution has become a serious problem for local governments in many cities in China. To tackle the air pollution issue, local governments have enacted many regulations to reduce auto exhausts, because vehicles are one major pollution source (Bektaş and Laporte, 2011, Franceschetti et al., 2013, Koç et al., 2014). One of the regulations is to allow only a small portion of vehicles that satisfy certain criteria (e.g., the license plate number ends with certain digits, local registered vehicles) to run in the city during the rush hours. For those vehicles not satisfying the criteria, their working times have to be divided into different segments caused by the rush hour regulation. This is very similar to the situation encountered by the inspector’s scheduling in the MPISP.
The segmentation of routing time makes the MPISP complicated and challenging. To tackle this problem, we first build a set-packing model and further strengthen it by the subset-row inequalities. On this basis, we propose a branch-and-price algorithm to solve this problem exactly, seeking for an optimal set of routes for inspectors to perform inspection jobs. The branch-and-price algorithm is operated on a branch-and-bound tree, and each node of the tree corresponds to a linear programming (LP) relaxation of the set-packing model, which is solved by a column generation procedure. The key in the column generation procedure is the pricing subproblem that identifies columns to enter into the basis of the LP relaxation. In our case, the pricing subproblem is a variant of the elementary shortest path problem with resource constraints (Feillet et al., 2004, Lozano et al., 2015) and is complicated by the working period constraints; and we design a tailored label-setting algorithm for it. Besides, we also present a tabu search procedure to solve the pricing subproblem heuristically, attempting to speed up the convergence of column generation.
The main contributions made in this paper are summarized as follows.
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First, to the best of our knowledge, this is the first work in the literature to present an exact algorithm for the MPISP. We build a set-packing model that is strengthened by subset-row inequalities, based on which we propose a branch-and-price algorithm to find optimal solutions.
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Second, to solve the pricing subproblem, which is a critical subproblem to be frequently solved in the branch-and-price, we design a tailored label-setting algorithm and moreover, accelerate it by bounded bidirectional search, label pruning, and decremental search space relaxation.
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Third, the computational results of the benchmark instances show the effectiveness of the proposed algorithm — two-thirds of the instances are optimally solved in a time limit of three hours per instance. The optimal solutions obtained in this study can serve as a convincing reference for future research for the MPISP and other related problems.
The remainder of this paper is organized as follows. We first discuss work related to our study in the next section. Section 3 describes the MPISP, presents the set-packing model, and introduces the shortest transit time function. Subsequently, in Section 4, we present a column generation procedure for solving the LP relaxation of the set-packing model, and elaborately describe the tailored label-setting algorithm for the pricing subproblem. The remaining components of the branch-and-price, including the initial solution, search strategy, and branching rule, are discussed in Section 5. Acceleration techniques are detailed in Section 6. We report the computational experiments and results in Section 7, and finally conclude this paper in Section 8. Note that all proofs are provided as Appendix A.
Section snippets
Related work
The MPISP was first introduced by Qin et al. (2015). They presented a tabu search algorithm to solve the problem heuristically, and formulated a constrained knapsack model to generate an upper bound. Compared with their work, our study is more accurate in the sense that we are directly pursuing an optimal solution for the MPISP by an exact algorithm.
The MPISP can be viewed as a generalization of the multi-period vehicle routing problem with profit (MPVRPP), which was first introduced by Zhang
Problem description and model
The MPISP is defined on a directed graph , where is the node set and is the arc set. Node 0 represents the depot where the inspectors are initially deployed, and the subset represents the set of n suppliers. Each arc has a travel time , and we assume that the travel time matrix satisfies the triangle inequality. Each supplier has an inspection task required to be fulfilled by an inspector and characterized by a
Column generation
Column generation, as an iterative procedure, is used to solve the LP relaxation of the set-packing model, which involves a huge number of variables. In each iteration, a restricted linear master problem (RLMP), which has the same constraints as the original problem but contains only a small portion of columns, is first solved with primal and dual solutions available. Then a subproblem, called the pricing subproblem, is solved to find new columns with positive reduced costs. If such columns
Branch-and-price algorithm
To solve the MPISP optimally, we propose a branch-and-price algorithm, which is a state-of-the-art algorithm adopted in routing problems (Baldacci et al., 2011, Luo et al., 2019, Luo et al., 2017). The upper bound (our problem is a maximization problem) at each node of the branch-and-bound tree is obtained by using the column generation procedure to solve the LP relaxation of the set-packing model.
Acceleration techniques
We have implemented four techniques, namely, bounded bidirectional search, label pruning, decremental search space relaxation, and tabu search column generator. The first threes are intended to accelerate the label-setting algorithm, while the last is a heuristic to solve the pricing subproblem.
Computational experiments
Extensive computational experiments are conducted on a set of benchmark instances to evaluate our exact algorithm. The algorithm was coded in Java, with ILOG CPLEX 12.5 for solving the restricted master problem. All the experiments were implemented on a PC equipped with an Intel i7-8700 CPU clocked at 3.20 GHz and 64 GB RAM, running on a 64-bit Windows 10 operating system. The time limit for the algorithm in solving an instance was set to 3 h (hour) in the experiments.
Conclusions
In this paper, we study the MPISP, which originated in the practice of product procurement for an international retailer. In this problem, we aim to find a set of routes for a team of inspectors that achieves the maximum total workloads, and the work of an inspector is to perform inspection jobs on several suppliers with different locations across a time horizon of multiple days. Since an inspector only works/travels during working hours and rests at other times, we divide the route for an
CRediT authorship contribution statement
Huaxiao Shen: Conceptualization, Methodology, Writing - review & editing, Validation, Funding acquisition. Shengnan Shu: Formal analysis, Methodology, Software, Writing - original draft. Hu Qin: Conceptualization, Data curation, Investigation, Writing - review & editing, Funding acquisition. Qinghua Wu: Supervision, Resources, Project administration, Funding acquisition.
Acknowledgments
This research was supported by Humanities and Social Science Fund of Ministry of Education of China under project No. 17YJC630120, and National Natural Science Foundation of China (NSFC) under project No. 71801231, No. 71771099, No. 71571077, No. 71971090, No. 71531009.
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